with \(\sum _{\vec{x}}\)  for a summation over the sites on a time slice,
and \(\hat{\Omega }_k\) 's are bent Polyakov loops with \(N_t +2\)  steps.
See Refs. [1]}, [2]} for the definition of \(\hat{W}_{\rm rec}\)  etc. and \(\hat{\Omega }_k\) .
